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%%文档的题目、作者与日期
\author{王立庆（2020级数学与应用数学1班） }
\title{随机分析入门习题解答 -- 条件期望}
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%\date{2021 年 9 月 14 日}
%\date{March 9, 2021}

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\maketitle

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\begin{enumerate}\itemsep1em

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\item  %第9题 1.4. Conditional Expectation
Find the correct statements about the conditional probability of the event $A$ given the event $B$. 
\begin{enumerate}
\item[a.]  The definition of conditional probability is $P(A|B) = \frac{P(A\cap B)}{P(B)}$. 
\item[b.]  $P(A|B)=P(A)$ if and only if $A$ and $B$ are independent. 
\item[c.]  In the definition of conditional probability, it is crucial that $P(B)$ is positive. 
\item[d.]  The probability measures $P(\cdot)$ and $P(\cdot | B)$ are always two different measures on the sample space $\Omega$. 
\end{enumerate} 

\vspace{0.2cm}

{\color{red}解答：abc. 在事件 $A$ 与 $B$ 相互独立的时候，概率测度 $P(\cdot)$ 与 $P(\cdot | B)$ 是一样的。在事件 $A$ 与 $B$ 不独立的时候，这两个概率测度是不一样的。

}

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\item  %第10题 
Roll a die and record the number. Let $B$ be the event that the number is neither one nor two. 
Let $F(x | B)$ be the conditional distribution function given the event $B$. Find $F(5 | B)$. 
\begin{enumerate}
\item[a.]  $5/6$.
\item[b.]  $2/5$.
\item[c.]  $3/4$.
\item[d.]  $2/3$. 
\end{enumerate} 

\vspace{0.2cm}

{\color{red}解答：c. 条件分布函数的定义为 $F(x | B) = \frac{P(X\le x, B)}{P(B)}$. 现在事件 $B=\{3,4,5,6\}$. 因此
$$P(5 | B) = \frac{P(X\le 5, B)}{P(B)} = \frac{P\{3,4,5\}}{P\{3,4,5,6\}} = \frac{3/6}{4/6} = \frac{3}{4}. $$ 

}

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\item  %第11题 
Roll a die and let a random variable $X$ record the number. Let $B$ be the event that $X\ge 3$. 
Let $\mathbb{E}(X | B)$ be the conditional expectation given the event $B$. Find $\mathbb{E}(X | B)$. 

\begin{enumerate}
\item[a.]  $7/2$.
\item[b.]  $4$.
\item[c.]  $9/2$.
\item[d.]  $5$.
\end{enumerate} 

\vspace{0.2cm}

{\color{red}解答：c. 随机变量 $X$ 在事件 $B$ 发生的条件下的条件期望的定义是 $$\mathbb{E}(X | B) = \frac{\mathbb{E}(XI_B)}{P(B)},$$ 其中事件 $B$ 的指示函数$I_B$ 当 $\omega\in B$ 时 $I_B(\omega) = 1$, 当 $\omega\notin B$ 时 $I_B(\omega) = 0$. 随机变量 $X$ 与 $XI_B$ 的分布律为
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|} \hline 
$X$ &1&2&3&4&5&6 \\ \hline 
$XI_B$ &0&0&3&4&5&6 \\ \hline 
概率 &1/6&1/6&1/6&1/6&1/6&1/6 \\ \hline
\end{tabular}
\end{table}

因此 $XI_B$ 的数学期望与 $(X | B)$ 的条件期望分别为
\begin{eqnarray*}
\mathbb{E}(XI_B) &=& 3\cdot\frac{1}{6} + 4\cdot\frac{1}{6} + 5\cdot\frac{1}{6} + 6\cdot\frac{1}{6} = 3, \\
\mathbb{E}(X | B) &=& \frac{\mathbb{E}(XI_B)}{P(B)} = \frac{3}{4/6} = \frac{9}{2}. 
\end{eqnarray*}
我们发现这个期望值是 3,4,5,6 的平均值。

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\item  %第12题 
Consider a random variable $X(\omega)=\omega$ on the space $\Omega = [0,10]$ with uniform distribution. 
Consider an event $B=(4,8]$. Find the conditional expectation $\mathbb{E}(X | B)$. 
\begin{enumerate}
\item[a.]  $2$.
\item[b.]  $4$.
\item[c.]  $6$.
\item[d.]  $8$.
\end{enumerate} 

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{\color{red}解答：c. 这里随机变量 $X$ 是连续型的。因为样本空间 $\Omega$ 上的概率分布是均匀分布，所以 $P(B)=\frac{2}{5}$. 现在计算 $\mathbb{E}(XI_B)$. 注意到 $X$ 的密度函数为 $f(x)=\frac{1}{10},\,\, 0\le x\le 10$, 所以
\begin{eqnarray*}
\mathbb{E}(XI_B) = \int_B xf(x)dx = \int_4^8 x\cdot \frac{1}{10}dx = \frac{12}{5}. 
\end{eqnarray*}
因此所求的条件期望为
\begin{eqnarray*}
\mathbb{E}(X I B) =  \frac{\mathbb{E}(XI_B)}{P(B)} = \frac{12/5}{2/5} = 6. 
\end{eqnarray*}

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\item  %第13题：多项选择
Consider a continuous random variable $X(\omega)=\omega$ on the space $\Omega = [0,10]$ with uniform distribution. 
Consider a discrete random variable $Y$ defined on the space $\Omega$ by 
\begin{eqnarray*}
Y(\omega) = \left\{ \begin{array}{ll}
100, & 0\le \omega\le 3, \\
200, & 3< \omega\le 6, \\
300, & 6< \omega\le 10.
\end{array}\right.
\end{eqnarray*}
Then the conditional expectation $\mathbb{E}(X|Y)$ is a discrete random variable with the following distribution. 
Find the values of $z_3$ and $p_3$. 

\begin{table}[ht]
\centering
\begin{tabular}{|p{2cm}|p{2cm}|p{2cm}|p{2cm}|} \hline
$\omega$ & $0\le \omega\le 3$ & $3\le \omega\le 6$ & $6\le \omega\le 10$ \\ \hline
$\mathbb{E}(X|Y)(\omega)$ & $z_1$ & $z_2$ & $z_3$ \\ \hline
概率 & $p_1$ & $p_2$ & $p_3$ \\ \hline
\end{tabular}
\end{table}

\begin{enumerate}
\item[a.]  $z_3=6$,\,\,\, $p_3=0.3$.  
\item[b.]  $z_3=7$,\,\,\, $p_3=0.3$.  
\item[c.]  $z_3=8$,\,\,\, $p_3=0.4$.  
\item[d.]  $z_3=9$,\,\,\, $p_3=0.4$.  
\end{enumerate} 

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{\color{red}解答：c. 离散型随机变量 $Y$ 将样本空间 $\Omega$ 分成互不相交的三个子集 $A_1=[0,3]$, $A_2=(3,6]$ 与 $A_3=(6,10]$. 
随机变量 $X$ 的概率密度函数为 $f(x)=1/10, 0\le x\le 10$. 按定义，当 $\omega\in A_3$ 时，
\begin{eqnarray*}
\mathbb{E}(X|Y)(\omega) = \frac{1}{P(A_3)} \int_{A_3} xf(x)dx = \frac{1}{4/10}\int_6^{10}x\cdot\frac{1}{10}dx = 8. 
\end{eqnarray*}

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\item  %第14题：多项选择
Let $X$ and $Y$ be random variables on the space $\Omega$. Let $\mathbb{E}(X|Y)$ be the conditional expectation. Find the correct statements. 
\begin{enumerate}
\item[a.]  $Z$ is a discrete random variable when $Y$ is a discrete random variable. 
\item[b.]  It has the property $\mathbb{E}[\mathbb{E}(X|Y)] = \mathbb{E}(X)$. 
\item[c.]  The conditional expectation $\mathbb{E}(X|Y)$ is a coarser version of $X$, so it is a function of $X$. 
\item[d.]  If $Y$ is a constant, then $\mathbb{E}(X|Y)=\mathbb{E}(X)$.
\end{enumerate} 

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{\color{red}解答：abd. 条件期望 $\mathbb{E}(X|Y)$ 确实是随机变量 $X$ 的一个粗糙的版本，但它不是 $X$ 的函数。
它将随机变量 $Y:\Omega\to\mathbb{R}$ 变成另一个随机变量 $\mathbb{E}(X|Y):\Omega\to\mathbb{R}$, 它是随机变量 $Y$ 的函数。
不同的随机变量 $X$ 给出了不同的函数。

}

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\item  %第15题：多项选择
Consider a sample space $\Omega$. Let $\mathcal{F}$ be a $\sigma$-field on $\Omega$. Find the correct statements. 
\begin{enumerate}
\item[a.]  The $\sigma$-field $\mathcal{F}$ is a collection of subsets of $\Omega$. 
\item[b.]  The $\sigma$-field $\mathcal{F}$ is not empty since the empty set $\varnothing$ and the set $\Omega$ are always in the $\sigma$-field. 
\item[c.]  If a subset $A$ is in the $\sigma$-field $\mathcal{F}$, then its complement $A^c$ is also in $\mathcal{F}$. 
\item[d.]  The union or intersection of a countable subsets in $\mathcal{F}$ are always in $\mathcal{F}$.
\end{enumerate} 

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{\color{red}解答：abcd. 这些都是 $\sigma$-域的定义中所要求的。


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\item  %第16题：多项选择
Let $\mathcal{F}$ be a $\sigma$-field on the sample space $\Omega$. Let $A,B\in \mathcal{F}$. Find the subsets that belong to $\mathcal{F}$.  
\begin{enumerate}
\item[a.]  $A\cap B$.
\item[b.]  $A\cup B$.
\item[c.]  $A-B$.
\item[d.]  $A\times B$. 
\end{enumerate} 

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{\color{red}解答：abc. 笛卡尔乘积集合 $A\times B$ 落在 $\Omega\times\Omega$ 中，超出了样本空间 $\Omega$ 的考虑范围。

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\item  %第17题：多项选择
Roll a die and let the random variable $X$ be the number observed. Let $\sigma(X)$ be the $\sigma$-field generated by $X$. 
Find the number of elements in $\sigma(X)$. 
\begin{enumerate}
\item[a.]  6.
\item[b.]  16.
\item[c.]  36.
\item[d.]  64.
\end{enumerate} 

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{\color{red}解答：d. 随机变量 $X$ 的可能取值为 $1,2,3,4,5,6$. 因此样本空间为 $\Omega=\{1,2,3,4,5,6\}$. 对每个 $1\le k\le 6$, 事件 $\{X=k\}$ 表示出现数字 $k$, 因此这些事件都在随机变量 $X$ 生成的 $\sigma$-域 $\sigma(X)$ 里。因为 $\sigma$-域在补集、可数并集、可数交集这些运算下是封闭的，所以 $\Omega$ 的每个子集都是 $\sigma(X)$ 中的元素。一共有 $2^6=64$ 个元素。

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\item  %第18题：多项选择
Find the correct statements about the Borel sets on $\mathbb{R}$. 
\begin{enumerate}
\item[a.]  The intervals $(a,b]$ are Borel sets. 
\item[b.]  The Borel $\sigma$-field $\mathcal{B}(\mathbb{R})$ is the $\sigma$-field generated by intervals of the form $(a,b]$. 
\item[c.]  Each subset of $\mathbb{R}$ is a Borel set. 
\item[d.]  Let $X:\Omega\to\mathbb{R}$ be a random variable, and $A\in\mathcal{B}(\mathbb{R})$ a Borel set. Then $X^{-1}(A)\in\sigma(X)$. 
\end{enumerate} 

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{\color{red}解答：abd. 选项 a 和 b 是 Borel 集的定义。并不是实数集的每个子集都是 Borel 集，因此选项 c 不对。选项 d 是随机变量的数学定义。对任意两个实数 $a<b$, 事件 $\{\omega\in \Omega\mid X(\omega)\in (a,b]\}$ 是这个随机变量所带来的一个事件，因此必须在这个随机变量生成的事件域 $\sigma(X)$ 里。

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\item  %第19题：多项选择
Let $B=\{B_t, t\ge 0\}$ be the standard Brownian motion. Let $\mathcal{F}_t$ be the $\sigma$-field generated by $(B_s, 0\le s\le t)$. Find the correct statements. 
\begin{enumerate}
\item[a.]  The $\sigma$-field $\mathcal{F}_t$ is the smallest $\sigma$-field containing the essential information about the stochastic process $B$ up to time $t$. 
\item[b.]  For any $0\le s\le t$ and any two real numbers $a<b$, the subset $A_s(a,b)=\{\omega\mid B_s(\omega)\in (a,b]\}$ is an element in $\mathcal{F}_t$. 
\item[c.]  For any $0\le s_1<s_2\le t$ and any Borel set $C\in\mathcal{B}(\mathbb{R}^2)$, the subset $A_{s_1,s_2}(C)=\{\omega\mid (B_{s_1}(\omega), B_{s_2}(\omega))\in C\}$ is an element in $\mathcal{F}_t$. 
\item[d.]  For any $0\le s_1<s_2<s_3$ and any Borel set $C\in\mathcal{B}(\mathbb{R}^3)$, the subset $A_{s_1,s_2,s_3}(C)=\{\omega\mid (B_{s_1}(\omega), B_{s_2}(\omega), B_{s_3}(\omega))\in C\}$ is an element in $\mathcal{F}_t$. 
\end{enumerate} 

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{\color{red}解答：abc. 根据时间的先后，由一个随机过程 $X = \{X_t,t\ge 0\}$ 可以生成的 一系列的 $\sigma$-域 $\{\mathcal{F}_t, t\ge 0\}$, 其中 $\mathcal{F}_t$ 是到时刻 $t$ 为止可能发生的样本路径全体。选项 d 中的三个时刻 $s_1 < s_2 < s_3$ 没有说明是否在时刻 $t$ 或之前，因此这个事件 $A_{s_1,s_2,s_3}(C)$ 不一定落在 $\mathcal{F}_t$ 中。任取时间区间 $[0,t]$ 中的有限个时间点 $s_1 < s_2 < \cdots < s_n$, 任取 $n$-维 Borel 子集 $C$, 事件 $A_{s_1,\cdots, s_n}(C)=\{\omega\mid (B_{s_1}(\omega), \cdots, B_{s_n}(\omega))\in C\}$ 的集合生成了这个 $\sigma$-域 $\mathcal{F}_t$. 

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\item  %第20题：多项选择
Let $X$ be a random variable and $\mathcal{F}$ a $\sigma$-field. Let $Z=\mathbb{E}(X|\mathcal{F})$ be conditional expectation. Find the correct statements. 
\begin{enumerate}
\item[a.]  The random variable $Z$ does not contain more information than that is contained in $\mathcal{F}$. 
\item[b.]  For all $A\in \mathcal{F}$, the random variable $Z$ satisfies the relation $\mathbb{E}(XI_A) = \mathbb{E}(ZI_A)$. 
\item[c.]  The random variable $Z$ is a coarser version of the original random variable $X$. 
\item[d.]  The conditional expectation $\mathbb{E}(X|Y)$ is defined by $\mathbb{E}(X|\sigma(Y))$. 
\end{enumerate} 

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{\color{red}解答：abcd. 选项 a 与 b 是条件期望的定义中的两个条件。选项 c 是说一个随机变量关于某个事件域的条件期望是一个“分辨率较低”的随机变量。一个例子，随机变量 $X$ 给出全班每个学生的成绩，事件域对应于班级的一些小组。则条件期望仅给出这些小组的平均成绩。而绝对期望则给出了整个班级的平均成绩。选项 d 是一个随机变量关于另一个随机变量的条件期望的定义。

}

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\end{enumerate}

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